Right triangle solver using feedback



1959 l. A. GREENWOOD, JR 2,872,112

RIGHT TRIANGLE SOLVER USING FEEDBACK Filed Feb. 28, 1956 /a I "ll" //vpurs j 6/ INVEN TOR. l/A/V A. GREAE/VWOOD, J/Q.

Unit-ed fit-ates Patent T RIGHT TRIANGLE SOLVER USING FEEDBACK Ivan A. Greenwood, Jr., Stamford, Conn., assignor to General Precision Laboratory Incorporated, a corporation of New York Application February 28, 1956, Serial No. 568,210

2 Claims. (Cl. 235-61) This invention relates to right triangle solvers and more specifically to right triangle solvers of the electrical analogue type in which the two sides of a right triangle are given and it is required to find the hypotenuse and an angle. Such a solver may also be termed a coordinate converter for, if the magnitudes of the two sides of a right triangle are considered to be the rectangular coordinates of a point, the hypotenuse and an angle constitute the polar coordinates of that point. That portion of the solver which solves for the angle is sometimes termed an arc tangent solver.

This invention provides the instrumentation for such right triangle solution with outputs in the form of shaft angular deflections, and is distinguished by the elimination of the need which conventional solvers have for accurate driver amplifiers. This invention also eliminates any necessity for employing linear synchro resolvers to insure accuracy.

The purpose then of this invention is to provide improved instrumentation for finding the hypotenuse and one angle of a right angle triangle when the two sides are given.

More specifically, the purpose of this invention is to provide a circuit employing components of low individual accuracy to secure a computed result of relatively high accuracy.

A further understanding of this invention may be secured from the following detailed description and the associated drawings, in which:

Figures 1 and 2 illustrate the operation of the invention.

Figure 3 is a schematic electromechanical diagram of the apparatus of the invention.

Referring now to Fig. 1, e and e represent any two vector quantities having orthogonal relationship. It is desired to find the magnitude E and direction 0 of their vector sum. The triangle OPA represents geometrically the relationships of these quantities, and trigonometrically they are expressed by geometric principles of e =E cos 0 e =E sin 0 6=tan (3) The value of E is given by Fatented Feb. 3, 1%59 2 hypotenuse E is the vector sum of the sides e and e and,

e,,=E cos 0' (5) e,,'=E' sin 0' (6) In the right triangle P'PC the hypotenuse E" is the vector sum of the sides D and D so that E =W m One instrumentation for performing the mathematical operations depicted in Fig. 2 is indicated in Fig. 3. Two input signals, e and e are received at conductors 11 and 12 respectively. These voltages may be either direct or alternating, with appropriate circuit components. However, 400 C. P. S. alternating voltage is employed in this example. These input signals are applied to two similar subtracting circuits 13 and 14 which may be of any type such as, for example, amplifiers with inputs including transformers having subtracting windings or resistors arranged in a subtracting network. Internally generated signals e and (2,. are also applied to the subtracting circuits, so that in circuit 13 the operation m" :n .c is performed, and in circuit 14 the operation e e '=D (9) is performed. The difference or error signals, D and D are amplified by amplifiers 16 and 17. These amplifiers need not be linear but Zero output should correspond to zero input since they amplify only error voltages and not components of the output signal voltages. The amplifier outputs are applied to the orthogonal field Windings 18 and 19 of a synchro resolver 21 having two armature windings 22 and 23 on its rotor 24 which set up a composite magnetic field representative of their vector sum E. If the strength of that magnetic field be termed F and its direction 0", then in which k is the constant of proportionality. When the rotor is positioned through its shaft 26 at an angle 0' voltages e and e will be induced in the output windings 22 and 23, having the values in which m is the constant of proportionality. These equations show that the device 21 is a coordinate transformer or coordinate rotator with equal windings, and reproduces the input orthogonal vector quantities in. a rotated coordinate system.

Substituting the value of F from (10):

The motor shaft 34 drives an gear 37; the angular position 0 output shaft 36 through of this shaft therefore is 0 :16, {S it 1 k e dt 17 and motor 42. The motor output shaft 43 has a speed S proportional to the input voltage 2 or in which C is a constant. Motor shaft 43 drives the E output shaft 44 at a speed proportional to S through the gear box 46 to a final angular displacement representative of the quantity E, or

k being the gear ratio. Substituting in Equations 13 and 14 for E" its value taken from Equation 7 e mkslDJ-l-D, sin (0) (23) e =mkVD, +D,, cos (00) (24) Equating Equations 18 and 23,

t mlh/DJ-FD] sin o"-e' 25 Equating Equations 22 and 24,

I t mkJDJ-l-D} cos e -a I (26) These equations show that in order for the shafts 36 and 44 to be servoed to stationary or null positions, both D, and D must be made equal to zero individually. When they are so made,

020' d dE' at a? both become zero, that is, the shafts 36 and 44 become stationary, and from Fig. 2 it is apparent that, when D and D become zero, e; becomes equal to e 2,. becomes equal to e 6 becomes equal to 0 and E becomes equal to E. The shafts 36 and 44 therefore b.- come the output signal shafts.

The quantities 0' and E should approach their final values 0 and E smoothly and directly. Described graphically, in Fig. 2 the point P should approach the point P along the straight path P'P. As so far described this may not occur, but wide excursions of Pfrom the straight path may be taken because, in Fig. 3, the angle 9' fed back to resolver 21 depends as so far described directly on 2 In general P will approach P by a spiral path. If less accuracy is required this is satisfactory, but for greater accuracy the straight path PP should be followed. This condition is obviously met if the rate of shrinkage of the side a; is proportional to the magnitude of 2 and also the rate of shrinkage of the side e is proportional to the magnitude of e That is, the desirable conditions are that E -I661 and de M, (28) Let ' =w a+ea Then, in the limit at null, e=E" and E'=E, so that approximately 1=l 'I( Difierentiating, considering [E'[ to be constant,

de I d 0l HT di (30) 1 d81 E F ?E dz (31) substituting in (27) d9' K61 a??? (32) That is, in order to secure the condition indicated by Equation 27, in the process of securing 0' the voltage representing e1 must be divided by a quantity representing E. a

It will be noted that this Equation 32 is in the same form as Equation 18, and that these two equations become identical when 51 V E'? The voltage divider 32 is provided to perform the operation of division indicated by the right side of Equation 32, dividing 2 by a function of E in the servomechanism 29 in order to secure the desired linear approach of the triangle solver to its null position. As is well known, a voltage divider inthe negative feedback path of a servomechanism divides the servomechanism input function by the function of the voltage divider. This is what is done by the divider 32, which is positioned proportionally to E, and which therefore .causes the input to amplifier 2S, and therefore its output,

This is what is demanded by Equation 32,

being proportional to the output speed of motor 31. The shafts 36 and 44 are made to change e and e until they equal e and e by connecting shaft 36 to a synchro resolver 51 having a single rotor coil 52 connected to the 400 C. P. S. source at the terminal 53. Two orthogonal stator coils 54 and 56 are provided, each loaded by a voltage divider 57 and 58 respectively. The sliders 59 and 61 thereof are connected to the conductors 62 and 63 respectively, and bear potentials e and (2,, which are applied to the respective subtracting circuits 13 and 14. The sliders 59. and 61 are connected for actuation by the E shaft 44. t

The synchro resolver 51 resolves the potential of its rotor coil 52 into the orthogonal potentials e and 2,," of its stator coils 54 and 56 so that e "-cos 0 (34) and t e -sin 6 These potentials are multiplied by E by the voltage and the values of eg' and e are progressively changed by the shafts 36 and 44 toward the values e 'and e which they attain at the null condition.

To recapitulate, inputs eg and e are subtracted from internally generated e and e, to form difference error signals D and D which eventually vanish. These error signals are amplified by amplifiers 16 and 17 which need not be accurate since at null, when the computation has been achieved, the amplifier outputs are zero. Their outputs during the automatic computation are converted by a double resolver and servomechanisms to shaft displacements approximating the desired output signals, which are converted by another synchro resolver to signals e and e approximating the input signals, and are balanced against them to secure the null signals.

The potential of terminal 53 in resolver 51 should preferably be from the same source a that employed in generating the input signals e and e thus eliminating any necessity for strict voltage regulation.

Linearity requirements are not significant in the resolver 21 or in the servomechanisms 29 and 39, since they carry only error signals which are zero at null, and not output signal components. It is therefore possible for each scrvomechanism to consist merely of a motor driven by a driver amplifier, dispensing with the tachometer generator and feedback loop.

What is claimed is:

l. A right triangle solver comprising, a pair of subtracting circuits each emitting an electrical difference signal, each of said pair of subtracting circuits having applied thereto a respective one of two electrical input min'uend signals representing two orthogonal vector quantities, a pair of voltage dividers each having a slider connected to a respective one of said pair of subtracting cirsuits for applying respective subtrahend signals thereto, a synchro resolver having an input winding energized by a constant potential and having a pair of output windings positioned in space quadrature, said two output windings being respectively connected to said two voltage dividers for applying multiplicant signals thereto, a coordinate transformer connected to said pair of subtracting circuits and actuated by the difference signals thereof to form transformed difference signals, first and second integrators connected to said coordinate transformer and having applied thereto said transformed difference signals producing therefrom respective angle and hypotenuse integral output signals, means for positioning said coordinate transformer and said synchro resolver by said angle integral signal, a dividing circuit associated with said first integrator, said dividing circuit being actuated by the hypotenuse integral output of said second integrator for dividing the value of the transformed difference signal applied to said first integrator by the value of said hypotenuse integral signal, and means for positioning the sliders of said two voltage dividers by said hypotenuse integral signal, whereby said subtrahend signals approach said minuend signals in value and said integral output signals approach the values of the vector sum and of the arc tangent of said two orthogonal vectors.

2. A right triangle solver comprising, a pair of electrical subtracting circuits each emitting an electrical difference signal and each having applied thereto a respective one of two alternating electrical input minuend signals representing by their voltage magnitudes two orthogonally related vector quantities, said two input signals having the same selected frequency and phase, a pair of linear voltage dividers each having a slider connected to a respective one of said pair of subtracting circuits for applying respective subtrahend signals thereto, a synchro resolver having a single input Winding energized by a constant alternating potential having said selected frequency and phase and having a pair of output windings positioned in space quadrature, said two output windings being respectively connected to said two voltage dividers for applying multiplicant signals thereto, a double synchro resolver having a pair of primary windings in space quadrature and a pair of secondary windings in space quadrature relatively rotatable by a shaft, amplifying means respectively connecting said pair of subtracting circuits to said pair of primary windings for actuation thereof by the amplified difference signals to form coordinate-transformed difference signals at said pair of secondary windings, a first integrating servomechanism including an amplifier, motor, generator and negative feedback loop and having a first output shaft, the angular displacement of said output shaft linearly representing the integral of input potential, said servomechanism being connected to one of said pair of secondary windings to form an angle output signal, a second integrating servomechanism including an amplifier, motor, generator and negative feedback loop and having a second output shaft, the angular displacement of said output shaftlinearly representing the integral of input potential, said servomechanism being connected to the other of said pair of secondary windings to form a hypotenuse output signal, said first output shaft being connected to said double synchro resolver shaft whereby said resolver shaft is positioned in terms of said angle output signal, said first output shaft being likewise connected to said synchro resolver for the rotational positioning thereof, a voltage divider associated with the negative feedback loop of said first integrating servomechanism, said voltage divider having a slider connected to said second output shaft and positioned in accordance with the angular displacement thereof whereby the value of the transformed dif ference signal applied from said one of the pair of secondary windings to said first integrating servomechanism is divided by a function of the value of said hypotenuse output signal, and a shaft connecting said second output shaft to the sliders of said pair of linear voltage dividers for positioning them in accordance with said hypotenuse output signal, whereby said subtrahend signals approach said minuend signals in value and said angle output signal and hypotenuse output signal respec tively approach the value of the arc tangent and of the vector sum of the orthogonally related vectors represented by said two alternating electrical minuent signals.

References Cited in the file of this patent UNITED STATES PATENTS 2,465,624 Agins Mar. 29, 1949 2,525,636 Bedford et a1. Oct. 10, 1950 2,569,328 Omberg Sept. 25, 1951 OTHER REFERENCES Servo Mechanism Practice (Ahrendt), page 1954 (McGraw-Hill) 

